Topology-Preserving Scaling in Data Augmentation

We propose an algorithmic framework for dataset normalization in data augmentation pipelines that preserves topological stability under non-uniform scaling transformations. Given a finite metric space \( X \subset \mathbb{R}^n \) with Euclidean distance \( d_X \), we consider scaling transformations defined by scaling factors \( s_1, s_2, \ldots, s_n > 0 \). Specifically, we define a scaling function \( S \) that maps each point \( x = (x_1, x_2, \ldots, x_n) \in X \) to \[ S(x) = (s_1 x_1, s_2 x_2, \ldots, s_n x_n). \] Our main result establishes that the bottleneck distance \( d_B(D, D_S) \) between the persistence diagrams \( D \) of \( X \) and \( D_S \) of \( S(X) \) satisfies: \[ d_B(D, D_S) \leq (s_{\max} - s_{\min}) \cdot \operatorname{diam}(X), \] where \( s_{\min} = \min_{1 \leq i \leq n} s_i \), \( s_{\max} = \max_{1 \leq i \leq n} s_i \), and \( \operatorname{diam}(X) \) is the diameter of \( X \). Based on this theoretical guarantee, we formulate an optimization problem to minimize the scaling variability \( \Delta_s = s_{\max} - s_{\min} \) under the constraint \( d_B(D, D_S) \leq \epsilon \), where \( \epsilon > 0 \) is a user-defined tolerance. We develop an algorithmic solution to this problem, ensuring that data augmentation via scaling transformations preserves essential topological features. We further extend our analysis to higher-dimensional homological features, alternative metrics such as the Wasserstein distance, and iterative or probabilistic scaling scenarios. Our contributions provide a rigorous mathematical framework for dataset normalization in data augmentation pipelines, ensuring that essential topological characteristics are maintained despite scaling transformations.

Machine Learning Algorithms to Assess Site Closure Time Frames for Soil and Groundwater Contamination

Monitored Natural Attenuation (MNA) is gaining prominence as an effective method for managing soil and groundwater contamination due to its cost-efficiency and minimal environmental disruption. Despite its benefits, MNA necessitates extensive groundwater monitoring to ensure that contaminant levels decrease to meet safety standards. This study expands the capabilities of PyLEnM, a Python package designed for long-term environmental monitoring, by incorporating new algorithms to enhance its predictive and analytical functionalities. We introduce methods to estimate the timeframe required for contaminants like Sr-90 and I-129 to reach regulatory safety standards using linear regression and to forecast future contaminant levels with the Bidirectional Long Short-Term Memory (Bi-LSTM) networks. Additionally, Random Forest regression is employed to identify factors influencing the time to reach safety standards. Our methods are illustrated using data from the Savannah River Site (SRS) F-Area, where preliminary findings reveal a notable downward trend in contaminant levels, with variability linked to initial concentrations and groundwater flow dynamics. The Bi-LSTM model effectively predicts contaminant concentrations for the next four years, demonstrating the potential of advanced time series analysis to improve MNA strategies and reduce reliance on manual groundwater sampling. The code, along with its usage instructions, validation, and requirements, is available at: https://github.com/csplevuanh/pylenm_extension.

How Analysis Can Teach Us the Optimal Way to Design Neural Operators

This paper presents a mathematics-informed approach to neural operator design, building upon the theoretical framework established in our prior work. By integrating rigorous mathematical analysis with practical design strategies, we aim to enhance the stability, convergence, generalization, and computational efficiency of neural operators. We revisit key theoretical insights, including stability in high dimensions, exponential convergence, and universality of neural operators. Based on these insights, we provide detailed design recommendations, each supported by mathematical proofs and citations. Our contributions offer a systematic methodology for developing next-gen neural operators with improved performance and reliability.

A Mathematical Analysis of Neural Operator Behaviors

Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous mathematical framework for analyzing the behaviors of neural operators, with a focus on their stability, convergence, clustering dynamics, universality, and generalization error. By proposing a list of novel theorems, we provide stability bounds in Sobolev spaces and demonstrate clustering in function space via gradient flow interpretation, guiding neural operator design and optimization. Based on these theoretical gurantees, we aim to offer clear and unified guidance in a single setting for the future design of neural operator-based methods.